DE102015005834A1 - Mixing techniques for curve fitting - Google Patents

Abstract

Mapping techniques for curve fitting are described. In one or more implementations, an indication is received over three or more data points. A blending factor is calculated based on a spatial relationship of the three or more data points to each other. A curve is fitted to the three or more data points by mixing a plurality of curve fitting techniques using the calculated blend factor.

Description

Related Application

The present application claims non-provisional priority to US Provisional Patent Application No. 61 / 992,083 filed May 12, 2014 and entitled "Blending Techniques For Curve Fitting", the entire disclosure of which is hereby incorporated by reference ,

background

Curve fitting involves fitting a curve (for example, a path) between data points. It can be used for a variety of different purposes, such as specifying the correspondence of data points to one another, as well as for tables, drawing animation paths, printing temperatures, and the like.

However, conventional techniques used for curve fitting may deviate from the expectations of the users who use the functionality. For example, unexpected extrema, loops, and the like may be observed in curve fitting using conventional techniques between those data points that do not follow the "flow" the user expects from the data points shown. As a result, users of conventional techniques are often forced to manually correct the curve, which can be frustrating and inefficient.

Summary

Mapping techniques for curve fitting are described. In one or more implementations, an indication is received over three or more data points. A blending factor is calculated based on a spatial relationship of the three or more data points to each other. A curve is fitted to the three or more data points by mixing a plurality of curve fitting techniques using the calculated blend factor.

In one or more implementations, a system includes one or more modules that are at least partially implemented in hardware. The one or more modules are configured to perform operations including: receiving an indication to fit a curve to a plurality of data points, determining a spatial relationship of the plurality of data points to each other, and mixing a plurality of curve fitting techniques to perform the fitting of the curve , wherein the mixing uses at least one heuristic based on the determined spatial relationship.

In one or more implementations, a computer program product is implemented, in particular, as a signal, data stream, and / or one or more computer-readable storage media, in particular, including instructions stored thereon that, in response to execution by a computing device, cause the computing device to perform operations that involve operations Selecting which of a plurality of curve fitting techniques to use based on a spatial relationship of a plurality of data points to which a curve is to be fitted and fitting the curve to the plurality of data points using the selected one of the plurality of curve fitting techniques.

In one or more implementations, a system includes one or more modules that are at least partially implemented in hardware, wherein the one or more modules are configured to perform operations that include: configuring a user interface to facilitate direct editing of complex curves Configuring contact points located directly on the complex curve to support user specified motion; and automatically and without user intervention fitting a respective complex curve curve such that the respective curve retains the slope and curvature continuity with respect to the complex curve.

The present summary introduces in simplified form a selection of concepts that are further described below in the Detailed Description. The summary as such should nor is it intended to be used as an aid in determining the scope of the claimed subject matter.

Brief description of the drawing

The detailed description is based on the accompanying figures. In the figures, the leftmost / leftmost digits of a reference numeral identify / identify the figure in which the numeral first appears. The use of the same reference numerals in various contexts of the description and in the figures may indicate similar or identical objects. Entities depicted in the figures may refer to one or more entities, and therefore reference to a single or multiple forms of the entities may be interchangeable in the explanation.

1 FIG. 10 is an illustration of an environment in an example implementation operable to employ the techniques described herein.

2 FIG. 12 shows in more detail a system in an exemplary implementation for representing a merge module of FIG 1 along with examples of curve fitting techniques that can be used by a curve fitting module.

3 FIG. 12 shows a usage example of a symmetry heuristic for generating a blending factor by the blending module to specify use of various curve fitting techniques.

4 shows an example of a rearrangement of a spatial relationship of one of the data points of 3 and a consequent change of a blend factor used to fit a curve to the data points.

5 FIG. 12 shows a usage example of flatness heuristics for generating a blend factor by the blend module for specifying use of various curve fitting techniques. FIG.

6 FIG. 12 shows a usage example of a sign heuristic for generating a blending factor by the blending module to specify use of various curve fitting techniques. FIG.

7 FIG. 14 shows an example of a construction of an example of a composite curve in which a longer κ curve is constructed by composing plural square Bezier segments with G1 continuity.

8th FIG. 10 shows an optimization scheme that may be formulated as shown in an example algorithm.

9 shows an example of a local behavior of a mode.

10 shows an exemplary system for solving κ curves containing up to 10,000 arbitrarily generated points.

11 Figure 14 shows an example of a comparison of a spline curve using the techniques described herein with a cubic Catmull-Rom spline interpolation technique.

12 FIG. 10 is a flow chart illustrating a procedure in an exemplary implementation that uses a blending factor to adjust a curve at three or more data points.

13 FIG. 10 is a flowchart illustrating a procedure in an example implementation in which a plurality of curve fitting techniques are selected for fitting a curve to a plurality of data points.

14 FIG. 12 shows an exemplary system with various components of an example device that may be implemented as a type of computing device as described and / or in connection with FIG 1 to 13 can be used to implement embodiments of the techniques described herein.

detailed description

Overview

Conventional techniques used to fit curves to data points may not imply sufficient intuitive interaction and ease of use. Further, such conventional techniques may also result in a curve that deviates from a user's expectations, for example, due to the occurrence of extremes, loops, and the like that do not occur in tracking the remaining data points, and therefore user expectations deviate when adjusting the curve.

Mapping techniques for curve fitting are described. In one or more implementations, blending techniques are employed to blend various curve fitting techniques to fit a curve to a plurality of data points, such as for tables, to plot animation paths, to print temperatures, and the like. For example, a computing device may receive an indication of successive data points by which to adjust a curve as a path between points. Heuristics are used to calculate a blend factor based on a spatial relationship of the data points to each other, such as symmetry, flatness, curvature, and the like.

This blending factor may then be used to select which of a plurality of curve fitting techniques to use to adjust the curve, as well as how to blend two or more of the techniques together. For example, the computing device may make an estimate using a quadratic Bézier curve fitting technique, for example, to estimate a tangent direction and a radius of curvature in the curve for the data points. These estimates may then serve as a basis for determining how to blend more curve fitting techniques together, such as a Bézier cubic curve fitting technique with a circular curve estimation technique. In this way, a mixture can be used for intuitively adjusting a curve, which will be further explained in connection with the following sections.

In the following explanations, an exemplary environment will first be described which may employ the mixing techniques described herein. Next, exemplary procedures that may be performed in the example environment as well as in other environments will be described. As a result, the performance of the example procedures is not limited to the example environment, and the example environment is not limited to the performance of the example procedures.

Exemplary environment

1 is a representation of an environment 100 in an example implementation operable to employ the techniques described herein. The illustrated environment 100 includes a computing device 102 which can be configured in a variety of ways.

The computing device 102 For example, it may be more configured as a desktop computer, laptop computer, mobile device (assuming a handheld configuration, such as a tablet or a mobile phone), and the like. As illustrated, for example, the computing device 102 configured in a mobile configuration as a tablet, which is a display device 104 having the functionality of a touch screen configured to detect touch points, such as those of the hand 106 of the user. This allows the computing device 102 from full-resource devices with significant memory and processor resources (e.g., personal computers, game consoles) to a low-resource device with limited memory and / or limited processing resources (e.g., mobile device). Although beyond that, a single computing device 102 shown, the computing device 102 represent a plurality of different devices, such as multiple servers used by a company to perform operations "over the cloud," as discussed below 14 is described.

The computing device 102 is shown as a curve fitting module 108 includes. The curve fitting module 108 provides functionality related to customizing one Curve to a plurality of data points 110 as well as functionality related to interacting (eg, modifying) the curve. Although the curve fitting module 108 as on the computing device 102 implemented, it may be implemented in a variety of ways, such as remotely via a service provider's web service, the "via cloud" over the network 112 is accessible between the service provider and the computing device 102 is distributed, and the like more.

In the example shown becomes a user interface 114 represented as having a plurality of data points 116 . 118 . 120 . 122 which are input via a gesture via the functionality of a touch-sensitive screen of the display device 104 is detected, for example, by using one or more capacitive sensors. This is done as part of a variety of different functionalities, such as specifying a path over which an object is to be animated, for specifying an edge in image processing, and the like. Other examples are also included, such as data points 110 received from a table, readings of a sensor, presentation software, and the like.

The curve fitting module 108 can then upon receipt of the data points 116 to 122 a curve 124 automatically and without user intervention based on the data points 116 to 122 to adjust. The curve fitting module 108 For example, it may be configured to have a plurality of different curve fitting techniques 126 quadratic Bézier curve techniques, cubic Bézier curve techniques and circular curve estimation techniques. Each of these curve fitting techniques 126 indicates given different spatial relationships of the data points 110 Strengths and weaknesses. Accordingly, the curve fitting module 108 a mixing module 128 which is a functionality for choosing which of the curve fitting techniques 116 for a given set of data points 110 should be used and how the majority of curve fitting techniques 126 should be mixed together to the final result of a curve 124 that is adapted to the points to reach. This allows the curve fitting module 108 be configured for a curve 124 which is visually pleasing and therefore by a user of the curve fitting module 108 "Expected", which will be further explained below and shown in the corresponding figure.

2 shows a system 200 an exemplary implementation for more detailed representation of the mixing module 128 from 1 along with examples of curve fitting techniques 126 that of the curve fitting module 108 can be used. The curve fitting module 108 is shown as having inputs for specifying data points 110 receives and a curve 124 that matches these data points. To perform this adjustment, the curve fitting module 108 a mixing module 128 which is a functionality for choosing which (and even "how many") curve fitting techniques 126 to adjust the curve 124 should be used.

To support this choice, the mixing module 128 a heuristic module 202 which is a functionality for employing one or more heuristics to initiate the fitting of the curve. For example, the heuristics may be based on a spatial relationship of the data points 110 to each other to adjust the curve, such as on a specified order of the data points 110 and a relationship of each of the data points to previous and / or subsequent regions of data points. Examples of such heuristics include symmetry 204 , a flatness 206 and a sign 208 the use of which will be further described in the following figures.

The heuristics from the calculation by the heuristic module 202 can then be used to generate a blend factor that can specify which of the curve fitting techniques 126 When adjusting the curve, you can specify a weighting that fits each of the curve fitting techniques 126 should be assigned, and the like more. Examples of such curve fitting techniques 126 include a square Bézier curve 210 , a cubic Bézier curve 212 , a circular curve estimate 214 and the like, which will be further explained in the following examples.

3 shows a usage example 300 a symmetry heuristic 204 for generating a mixing factor by the mixing module 128 for specifying a use of various of the curve fitting techniques 126 , First and second examples 302 . 304 of curves 306 . 308 with adaptation to the data points 310 . 312 . 314 . 316 are shown. In the first example 302 becomes a curve 306 to the data points 310 . 312 . 314 . 316 using the square Bézier curve 210 from 2 customized. tangents with controls or controls are at respective data points 310 to 316 for specifying scale factors for curve fitting between adjacent curves.

Drawn circles inside the curve 306 show a radius of curvature at respective data points 310 to 316 , Thus, as indicated by the circles, the curves that point between the data points 310 to 316 in the first example 302 are shown, curvatures that vary, ie, for example, decrease toward a center and then increase again when approaching another data point. Therefore, the curve can 306 deviate from the user's expectation of a circle.

In the second example 304 is a curve 308 also between the data points 310 to 316 however, in this case, a circular curve estimation technique 214 supports. The circular curve estimation technique 214 estimates a circle for each collection of three consecutive data points. Therefore, as shown, the radius of curvature follows at respective data points 310 to 316 (For example, inner circles) in addition to a circle through the curve 308 what can be used to achieve a desired result for a user, such as the likelihood that a circle is drawn in this example.

The heuristic module 202 of the mixing module 128 from 2 can use a variety of different heuristics to create a blending factor that can describe which of the curve fitting techniques 126 should be used, and even arrive at a weighting to be assigned to the various factors. As with the second example 204 For example, three data points may be shown 310 . 312 . 314 be defined in sequence. A baseline 318 can be between the first and third data points 310 . 314 be defined, as well as a relationship 320 of the second data point 312 (that is, the data point that is in succession between the other two data points 310 . 314 is defined) to the baseline 318 with use for defining the mixing factor. The relationship 320 For example, it can be defined based on a point that is approximately perpendicular to the baseline 318 to the second data point 312 is. This can be used to determine if the second data point 312 between the first and third data points 310 . 314 along the baseline 318 falls, if so, then the curve fitting technique 126 the circular curve estimate 214 that is, it will be given a higher weighting than the quadratic Bézier curve technique 210 assigned.

For example, weightings may be based on a closeness of the relationship 320 to a midpoint of the baseline 318 be used, for example, a greater weighting of the Kreiskurvenschätztechnik 214 is assigned the closer the midpoint is. Once you are "outside" the baseline 318 between the first and third data points 310 . 314 is, can be a greater weighting of another curve fitting technique 126 such as the square Bézier curve technique 210 , be assigned. An example of this is described below and shown in the corresponding figure.

4 shows an example 400 rearranging a spatial relationship of one of the data points 312 from 3 and a corresponding change of a blend factor used to fit a curve to the data points 310 to 316 , As shown, a cursor control device (eg, a mouse) moves the data point in this example 310 from an original place, as in 3 is shown to a new location, as shown by a dashed arrow.

This leads to a new symmetric relationship between the data points 310 . 312 . 314 to the effect that the relationship 402 of the second data point 312 to the baseline 318 with formation between the first and third data points 310 . 314 "Outside" the points, that is not between the data points 310 . 314 along the baseline 318 is arranged. Therefore, the mixing module 128 get to a blending factor, the square Bézier curve technique 210 a greater weight than the circular curve estimation technique 214 in this example, to fit a curve according to the probable intuition of the user. This allows mixing to be done based on how well the data points are 310 to 316 approximate a circle by weighting the various curve fitting techniques.

5 shows a usage example 500 a flat heuristic 206 for generating a mixing factor by the mixing module 128 for specifying the use of various curve fitting techniques 126 , In this example, the second data point becomes 312 towards the baseline 318 passing through the first and third data points 310 . 314 is defined, moved. This causes the curvature for the curve segments passing through the first, second, and third data points 310 to 312 and 312 to 314 is defined as "flattening out". For this purpose, the mixing module 128 the square Bézier curve 210 a bigger one Weighting, the closer the second data point 312 to the baseline 318 than the circular curve estimation technique 214 assign, for example, when a predefined threshold is exceeded. Thus, as easily from the explanation too 3 to 5 As can be seen, a plurality of heuristics are used simultaneously to provide a blending factor for a plurality of curve fitting techniques 126 to calculate.

6 shows a usage example 600 the sign heuristic 208 for generating a mixing factor by the mixing module 128 for specifying the use of various curve fitting techniques 126 , The example shown 600 includes first, second and third data points 602 . 604 . 606 to which a curve 608 to be adjusted. In this example, the merge module examines 128 a curvature sign on adjacent data points to help calculate the mix factor.

As for the first and second data points 602 . 604 is shown, for example, a curvature sign 602 for the first data point 602 from a curvature sign of the second data point 602 respectively. 604 different. In other words, the curves bend in opposite directions. The mixing module 128 Therefore, in this case, it can produce a mixing factor that does not use a circular-curve estimation technique 214 which is why another technique is used, such as the Bézier quadrature curve technique 210 is selected. In this way, out of line falling curves can be obtained by using the circular curve estimation technique 214 can be avoided, thereby causing the curve 604 in this example, from the data points 602 to 606 "Drifts away" (swoop away). Further examples of curve fitting techniques will be found in the following explanation.

In the following discussion, examples of heuristics and curve fitting techniques will be described in more detail.

Square Bézier κ curves

Consider a square Bézier curve of the following form: r (t) = (1-t) ² r ₀ + 2t (1-t) r _m + t ² r ₁ (1)

The function r (t) specifies the coordinate trajectory of the curve with parameterization by the time parameter t, whose value varies between 0 and 1. Note that at t = 0, the curve passes exactly through the control point r ₀ , while at t = 1 the curve passes through the control point r ₁ . However, the curve generally does not pass through the middle control point r _m .

Square interpolation of Bézier curves

A quadratic Bézier three-point interpolation scheme may be created in which the location of the point r _{c is} controlled to be on the curve at a position of maximum curvature. For example, for a set of data points r ₀ , r _c , r ₁ , the curve starts at r ₀ , passes through r _c, and ends at r ₁ . An arbitrary time value t _c ε (0, 1) can be selected and resolved for the mean Bézier control point. Here and for the rest of the explanation, the spelling (0, 1) is used to represent an open interval from 0 to 1, that is, an interval that does not include the endpoints.

Therefore, based on equation (1), the following expression can be written. r _c = (1 - t _c ) ² r ₀ + 2t _c (1 - t _c ) r _m + t 2 / cr ₁ (2)

A simple transformation of the terms of 2 results:

From equation 3 it can be seen that a solution always exists for values with t _c ε (0, 1). The degenerate cases of t _c equal to 0 or 1 are straight line segments.

arc length

The arc length s (t) of a curve is given by integrating the velocity of the curve over time as follows: s (t) = ∫ t / 0 | r '(u) | du (4)

Note that the primed spelling designates derivatives in terms of the time parameter.

maximum curvature

The curvature κ (t) of a continuous space curve is defined as the rate of change per unit arc length of the unit tangent vector along the curve. The unit tangent vector is given by: T (t) = r '(t) / | r' (t) | (5)

By definition, the following can be specified:

For 2D curves (2D two-dimensional), equation (6) can be converted to a simpler expression as follows:

Here, the operator designates the cross product operator (that is, given two given vectors p = [p _x , p _y ] and q = [q _x , q _y ], p x q = p _x q _y - p _y q _x ). The quadratic Bézier expression given in equation (1) can then be differentiated, yielding the following: r '(t) = v (t) = (2t - 2) r ₀ + 2 (1 - 2t) r _m + (2t) r ₁ (8) r '' (t) = a (t) = 2r ₀ - 4r _m + 2r ₁ (9)

In this case, the variables v (t) and a (t) are used to denote the speed or the acceleration of the curve. The denominator indicates the "velocity" of the curve at time t. Note that for a quadratic Bézier expression, the acceleration term is constant. At time t = 0 and t = 1, the velocities of the curve are then given by: v ₀ = 2 (r _m - r ₀ ) (10) v ₁ = 2 (r ₁ -r _m ) (11)

For a quadratic Bézier curve, the velocity varies linearly with t as follows: v (t) = v ₀ + ta ₀ (12)

These values can be included in the expression for the curvature (7), which gives the following:

Note that the numerator of expression (13) is constant. However, as described above, it is still desirable to place a control point at an approximate location where the curvature reaches its maximum value. This involves determining the time parameter for which the maximum curvature value is achieved. Since the counter in Expression (13) is constant, a maximum can be set function by minimizing the denominator. Equivalently, the square of the velocity can be minimized by differentiating it and setting it equal to 0: 0 = d / dt (v ₀ ⁺ t _c a ₀ ) ² = d / dt (v ₀ * v ₀ + 2t _c v ₀ * a ₀ + a ₀ * a ₀ t 2 / c) = 2 (v ₀ · a ₀ + a ₀ · a ₀ t _c ) (14)

By solving for t _c , a curve reaches its maximum curvature value at:

Maximum curvature Time

Assume that there are three points for interpolation, namely r ₀ , r _c , r ₁ , where r _{c is} exactly at the point of maximum curvature along the square Bézier curve. From equation (3) it follows that the curve interpolates the three points exactly for any time value t _c ε [0, 1]. Thus, a time value t _c ε [0, 1] can be searched for which the quadratic Bézier curve reaches a local maximum curvature at time t _c (that is, equation (15) is satisfied).

Using equation (10) and equation (3), the velocity vectors for the new curve can be expressed as follows:

Here, the writing r _{i, j is used} to denote the difference of positions, that is, for r _{i, j} = r _i -r _j . The acceleration can be calculated using equations (9) and (3) as follows:

This is constant along the curve. Finally, by combining equation (15) with the expressions for velocity (16) and acceleration (18), an expression can be determined for the time t _c at which the maximum curvature occurs in the Bézier quadratic expression:

Equation (19) is a cubic equation: r 2/3 1.0t / c - _c 3 _r, r ₀ · _1.0 t 2 / c + (2r 2 / c, 0 + r _{c, 0} · r _1.0) t _c - r 2 / c, 0 = 0 (20)

The solution can be done either by the cubic formula or by a numerical approach. For simplicity, a few iterations may be used by Halley's method, providing a cubic convergence to the solution. This gives a solution for a square Bézier three-point κ curve. In summary, when three arbitrary points r ₀ , r _c and r _{1 are} given, where r _{c is} the location of the maximum curvature, equation (20) can be solved to calculate the maximum curvature time t _c , whereupon a solution for the location of the middle Bézier control point r _m using equation (3).

Proof of a single solution

Note that the cubic equation (20) has a single real root within the time domain of interest 0 ≤ t ≤ 1. This is because, as indicated by (12), a quadratic Bézier curve follows a trajectory equivalent to that of an object moving at a constant acceleration, which is known as a parabolic path. As such, only a single maximum curve may be present. This can be proved directly by examining the essence of the roots of cubic equations, which can be readily seen by rewriting the equation in the following way: f (t) = t ³ - 3λ cos (θ) t ² + λ (2λ + cos (θ)) t - λ ² = 0 (21)

Here are λ = | r _{c, 0} | / | r _1,0 | (22) and cos (θ) = r _1.0 · r _{c, 0} / (| r _{c, 0} || r _1.0 | (23)

The values of f (t) at the two endpoints for t = 0 and t = 1 are: f (0) = -λ ² (24) and f (1) = 1 - 2λcos (θ) + λ ² = (1 - λ) ² + (2λsin (θ / 2)) ² (25)

By neglecting the trivial special case for λ = 0 and t _c = 0, it is stated that the function f (t) is strictly negative at t = 0, and it is neglecting the special case with λ = 1, λ = 0 and t _c = 1, indicated that this is strictly positive at t = 1. It follows that there is at least one real root lying between t = 0 and t = 1.

It can now be shown that only a single real root is present. The alternative situation with three real roots can be determined as follows. Assume that there are actually three real roots in the designated area. Then, the derivative function df (t) = dt or df (t) / dt must have two zeros or zeroes in that range, and the second derivative function d ² f (t) / dt ² must also disappear in the range. The latter condition implies: d ² f (t) / dt ² = 6 (t - λ cos (θ)) = 0 (26)

This in turn implies: 0 ≤ τ ≤ 1 (27)

Here, the variable t is defined as follows: τ = λ cos (θ) (28)

The requirement that the derivative function is 0, ie df (t) / dt = 3t ² - 6λ cos (θ) t + λ (2λ + cos (θ)) = 0 (29) produces two roots, namely:

The term below the square root can be rewritten as follows: - [τ (1 - τ) + 2λ ² sin ² (θ)] / 3 (31)

Neglecting the aforementioned special cases, this is negative, with the implication that there are no real roots of the derivative function in the desired range of t. Therefore, the assumption of three real roots is not valid, so there is only one single root of the cubic equation (21) within the desired range.

Note also that at λ> 0 the cubic expression can not have a root at t = 0. Thus, as long as r ₀ and r _{c are} different, t _{c is} different from 0. Similarly, when r ₁ and r _{2 are} different, t _{c is} different from 1 due to symmetry.

Composite square Bézier κ curves

The above discussion has detailed how square Bezier three-punk κ curves are constructed. In this section, the discussion of how to connect multiple square Bezier segments together to form continuous C1-κ cues. 7 shows an example 700 FIG. 5 shows the construction of an example of a composite curve in which a longer κ curve is built up by composing a plurality of square Bezier segments with G1 continuity.

For N maximum curvature points there are N square Bézier curve segments and N corresponding boundary condition equations. To simplify the following equations, the Bézier quadratic components are defined as follows:

Here, t _ci denotes the maximum curvature time for each of the N specific maximum curvature locations, each of which is indexed i.

For each quadratic Bézier curve, the coupled equations can be written as follows:

G1 continuity requires that the tangent directions and magnitudes (velocities) at the transition point between curve segments be continuous. This condition implies that the Bézier endpoints are midway between their neighboring middle control points:

To formulate the solution, the mean control points are solved, from which the solution for the Bézier end points can then be derived. Using equations 33 and 34, this approach to control points yields the following recurve relationship:

The boundary condition of the known boundary end points r ₀ and r _N-1 leads to the following equations at the edges:

hinsichtlich der bekannten Randbedingungsvektoren b gelöst werden kann. Die Matrix hat Dreieckform:

Together, this yields the matrix equation Ax _m = b, which for the unknown vectors

with regard to the known boundary condition vectors b can be solved. The matrix has triangle shape:

Note that with the exception of the first and last rows, the sum of all elements of a row is equal to 1, which follows from the Completeness Editor α _n + β _n + γ _n = 1. In addition, the diagonal elements which are equivalent to (1 + β _n ) / 2 are in a range of 1/2 to 3/4. The maximum value of 3/4 occurs when the corresponding maximum curvature time is 1/2. Furthermore, the matrix is diagonally dominant, and if none of the time parameters have the values 0 or 1, then the matrix is strictly diagonally dominant. In these cases, the matrix is known to be non-singular (det (A) ≠ 0) due to the Levy-Desplanques theorem.

Algorithm for quadratic κ-curves

The system described above is not linear in nature. Note that the construction of the matrix A and the right side vector b starts from knowing the maximum curvature times for each of the maximum curvature points. Calculating the maximum curvature times t _ci implies knowledge of the locations of the Bézier endpoints r _i that are the solutions to the matrix system. This non-linear dependence prevents the formulation of an analytical solution. Instead, an optimization scheme may be formulated as in the example algorithm 800 who in 8th is shown.

The analysis begins with an estimate of the initial positions r _{i of} the curve by averaging the adjacent maximum curvature points. The inner loop relates to updating estimates for the maximum crawl times, followed by updating the coordinate positions by solving the above-described matrix equation. The subroutine "computeMaxCurvatureTime" solves the cubic equation (20). An intuitive understanding is that time estimates can be improved and then used to create improved locations. These locations can then be used to recalculate a new set of time estimates, and so on, until the system reaches convergence. A variety of different convergence criteria may be used, such as when the time values t _ci remain nearly unchanged during successive iterations.

Locality for collinear boundary conditions

The above-described quadratic Bézier formulation implies a global solution. For such a solution, as the number of points on the curve increases, the corresponding calculation grows. Furthermore, the global nature of the solution suggests that local changes to the interpolation points have global effects on the shape of the curve. Such behavior may be undesirable from both a performance and a conceptual perspective in some cases. In practice, the solutions are found to be stable and local, with non-local effects disappearing as one moves further along the path from Bézier segment to Bézier segment. In this section, a theoretical explanation for these observations is provided by focusing on a particular set of points that exhibit oscillatory behavior in a particular case, but in a more common case supporting exponentially damped behavior.

This character of propagation of a change on the curve can be examined by considering the specific example of a curve formed by a set of collinear κ points, using the notation κ points to designate the maximum points of curvature. These points may be chosen such that they all lie along a horizontal line with the vertical axis coordinate value equal to zero. In the same way, the last curve end point is also limited. For example, imagine a long chain of maximum curvature points that looks at the behavior of a section that does not contain an endpoint. The κ points are then of the following form:

There is then given a solution of the following form:

Such a solution has an exponentially decreasing amplitude for a positive λ. For an ω equal to 1, the behavior is oscillatory. This oscillatory behavior is limited to a special arrangement of the κ-point curve. From (35) it follows that when subtracting the same equation with an index n decremented by 1 and applying the completeness relations, the following applies:

The following definitions are used here: δζ ζ _n = _n - ζ _n-1 (44) δη _n = η _n - η _n-1 (45)

The resulting oscillatory decreasing solution for n leads to the following form for δη _n : δη _n = η _n (1 + ω) / ω (46)

This is equivalent to δη _{n + 1} : δη _{n + 1} = -η _n (1 + ω) (47)

Thus: η _n = ½ (1 + ω) (α _n / ω + γ _n ) η _n (48)

This leads to the following condition: 1 = ½ (1 + ω) (α _n / ω + γ _n) (49)

This is true when there is an oscillatory or oscillatory decreasing solution to the collinear κ-point problem. For the particular oscillatory case where the magnitude of the vertical component of the Bézier control points oscillates above and below the line of constant amplitude κ points, the "decrease factor" ω is equal to 1, resulting in the following requirement: Oscillation: 1 = α _n + γ _n (50)

This is true for the two values of the corresponding maximum curvature times, that is for t = 0 and t = 1. The solution t = 0 is consistent with the cubic equation of time, which constrains the boundary conditions to points of maximum curvature. In this case, the cubic equation reduces to the following: -Λ ² = 0 (51)

This applies if the maximum curvature point lies exactly at the point of its end point with t = 0. In contrast, the solution t = 1 leads to the cubic equation: 1 - 2λcos (θ) + λ ² = 0 (52)

This has a real solution for λ only for the special case λ = 0 and θ = 0. In this case, the maximum curvature point lies exactly at the end point with t = 1.

The amplitude of the oscillation is limited by the properties of the square Bézier curve. Consider a square Bézier curve segment with the end point r ₀ , which can vary freely, whereas the maximum curvature control point r _c and the end point r _{1 are} both fixed. Without limiting the generality, the latter points can be placed on a horizontal line, with a spacing between them of d = r _{c, 1} = | r _{c, 1} | is available. The point r _c is placed at the origin of the coordinate system. The solution for the maximum vertical deviation of the curve section with position between r _c and r ₁ is given in the following form for the first end point r ₀ = (0, y ₀ ) with the time parameter t _c for the maximum curvature point by the following expression:

The vertical extent of the curve segment is given by:

This results in a linear scaling with the point spacing distance d. This expression is maximized by the choice y ₀ = 1, t _c = 1/2 and the maximum vertical deviation δy _max = (1/8) d. The maximum vertical amplitude η _{max of} the mean Bézier control point is given as η _max = (1/2) d. In other words, the "vertical jitter" of the maximum curve segment in a movement of the first endpoint is limited by one-eighth of the spacing d, the amplitude of which is a quarter of the maximum vertical amplitude of the Bézier control point. This is a result for a single square Bézier curve. When multiple curves are added to such a path, this maximum "jitter" is generally reduced, usually in an exponentially decreasing manner, as indicated in the collinear constraint point example.

Again, given the general oscillatory decreasing solution, after expressing α _n and γ _n explicitly as functions of the corresponding time parameters, the following equation relating t _n to ω is determined: 1 = ½ (1 + ω) ((1 - t _n ) ² ω + t 2 / n) (55)

This has the following solution: t _n = (1 - √ (Ω) ) / (1 + ω) (56)

The time parameter does not depend on the point index and is therefore constant along the curve, except for endpoints that are not included here. The requirement 0 ≤ t _n ≤ 1 restricts ω to the position between 0 and 1. The value of interest for t is t = 1/2, which for a value of ω = 3 - 2√ (2) = 0.171573 occurs. In this case, the Bézier control point vertical amplitude decreases exponentially as the dot index increases with approximately (0.171573 ...) ⁿ = exp (-1.76275n). In general, the closer t is to 0, the slower the rate of decrease, and the closer t is to 1, the greater this rate.

The horizontal components are now determined to understand the dependence of the solution on the point spacing. The focus here is on the situation of an equal horizontal spacing δζ of the mean Bézier curve control or control points: δζ _n = δζ _{n + 1} = δζ _n (57)

This leads to:

The last term describes the deviation of the horizontal position of the middle Bézier control point relative to the associated κ point thereof. The oscillatory solution with t _n = 0, ω = 1 therefore has the Bézier control points with spacing midway between the κ points, that is, the offset is half the κ point spacing.

The aforementioned special case for t _n = 1/2 does not lead to an offset. For the exponentially damped case, the middle Bézier control points lie at the same horizontal location as the maximum curvature boundary conditions.

Therefore, an exemplary curve is described which supports an exponential decrease in the curve amplitudes, as well as a pathological case of the constant limited amplitude oscillation. While the quadratic Bézier formulation leads to a global solution with the potential for nonlocal propagation of curve changes, in practice local curve changes are severely attenuated, such as with exponential decay with collinear arrangement of the points. As in example 900 from 9 can be seen, the visual behavior of the model is by nature comparatively local. A local change in the curve tends to affect the curve in a limited local area.

Exemplary implementation

A variety of different cubic approaches may be used to solve equation (20), such as a direct cubic approach such as the Halley method. In practice, Halley's method offers the best level of performance as well as the best stability. This is partly due to the ability to fill this method with an initially guessed value close to the true solution, as follows: t _c = | r _{c, 0} | / (| r _{c, 0} | + | r _{c, 1} |) (62)

The observation is that the interpolating quadratic expression closely follows the polyline that runs through r ₀ , r _c , r ₁ . For example, the initial rate value may even be exact if the three points are collinear. In one or more implementations, the iterations of Halley's method are performed until the change of t _c falls below 1.0e-6.

In order to solve the matrix system A, a basic implementation of a sequential triangular approach is used which has a known transit time of O (n) as well as a known memory of O (n). For the convergence criteria for algorithm 1 in example 800 from 8th For example, iterations are performed until the change in the calculated maximum curve times t _ci falls below 1.0e-6 for each square Bezier segment, for example.

In the example 1000 from 10 For example, an exemplary system for solving κ curves is shown having up to 10,000 arbitrarily generated points in the interval [0, 1]. Note that the overall solution time is in the sub-second range and that the power scaling is nearly linear in terms of the size of the curve due to the convergence rate and the efficiency of the chosen matrix approach.

11 shows an example 1100 a comparison of a spline curve using the techniques described herein with a cubic Catmull-Rom spline interpolation method [Catmull and Rom 1974]. The choice of parameterization may affect the quality of the results of the Catmull-Rom model. 11 FIG. 12 shows a configuration in which a default selection of a voltage behaves unfavorably and generates loops and shivers (dotted lines) between the control vertices, whereas the techniques described here produce smooth and intuitive results (shown in a solid line). Although a user can globally adjust the voltage parameters to avoid these loops and shivers in the conventional art, this reduces the fairness of the curve by concentrating a high curvature at the control vertices. A user may also separately adjust the voltage at each vertex to produce better results, but this implies manual tuning.

Aesthetic cubic G2-Bézier curves

The foregoing sections describe techniques that can be used to formulate continuous quadratic C1-κ curves. Although the solution for square κ curves has an elegant shape, it is often desirable to achieve higher order continuity. For example, for CAD applications, G2 continuity is usually considered a minimum requirement. For two-dimensional drawing applications, however, such continuity is not necessary. Furthermore, it is known that the smoothness of higher order does not always lead to more aesthetic curves.

In the case of two-dimensional character systems, the aesthetics and the controllability of the curve representation play a greater role than the overall continuity. Unfortunately, there is not a good known measure of fairness with which to evaluate curves, though proposals have been made for this, such as minimizing the overall variation in curvature. Minimizing curve variation results in overall rounded or circular curves that do not necessarily meet the expectations of an artist. Note that the square κ curves, although producing pleasing curves, may have a rapidly changing curvature. Therefore, a curve having higher order continuity (ie G2) can be tracked with less variation in curvature to produce a more visually pleasing curve model.

For this purpose, cubic Bézier curves can be used. Cubic Bézier curves are the basis for many curve modeling systems and are presented in standard rendering systems. There are tons of literature for efficiently rendering cubic Bézier curves on a graphics processing unit. However, as shown below, the mathematical requirements for such curves may be complicated in the analysis. To overcome these problems, techniques for solving the problem and demonstrating that under most conditions G2 curves can be generated which are both aesthetic and have a low curvature variation are described below, the maximum curvatures being close to the control and detection angles Control Vertices are located.

A cubic Bézier curve has the following form: r (t) = (1-t) ³ r ₀ + 3t (1-t) ² r _m0 + 3t ² (1-t) r _m1 + t ³ r ₁ (63)

The first and second derivations of this are as follows: υ (t) = -3 (t ² - 2t + 1) r ₀ + (9t ² - 12t + 3) r _m - (9t ² + t) r _n + 3t ² r ₁ (64) a (t) = 6 (1-t) r ₀ + 6 (3t-2) r _m + 6 (1-3t) r _n + 6tr ₁ (65) a '(t) = -6r ₀ - _m 12r - 12r _n + 6r ₁ (66)

The speeds and the acceleration at the beginning and the end can be expressed as follows: υ (0) = 3 (r _m - r ₀ ) (67) υ (1) = 3 (r ₁ -r _n ) (68) a (0) = 6 (r ₀ - 2 r _m + r _n ) (69) a (1) = 6 (r ₁ - 2 r _n + r _m ) (70)

The curvatures at t = 0 and t = 1 are as follows:

This is further simplified to:

Assume two points r ₀ and r ₁ given along unit longitudinal tangent directions T ₀ and T ₁ at the two points. In addition, assume certain curvatures κ ₀ and κ ₁ at r ₀ and r ₁ . Then the following two expressions can be used:

By simplifying these expressions, we obtain the following:

This is further simplified to: T ₀ × r _0n = 3 / 2κ ₀ λ 2/0 (79) T ₁ × r _1m = 3 / 2κ ₁ λ 2/1 (80)

It follows: T ₀ × (r ₀ - (r ₁ + λ ₁ T ₁ )) = 3 / 2κ ₀ λ 2/0 (81) T ₁ × (r ₁ - (r ₀ + λ ₀ T ₀ )) = 3 / 2κ ₁ λ 2/1 (82)

A multiplication provides: T ₀ × r ₀₁ - λ ₁ T ₀ × T ₁ = 3 / 2κ ₀ λ 2/0 (83) T ₁ × r ₁₀ -λ ₀ T ₁ × T ₀ = 3 / 2κ ₁ λ 2/1 (84)

Finally: (3 / 2κ ₀ ) λ 2/0 - (T ₀ × T ₁ ) λ ₁ - (T ₀ × r ₁₀ ) = 0 (85) (3 / 2κ ₁ ) λ 2/1 - (T ₀ × T ₁ ) λ ₀ - (T ₁ × r ₁₀ ) = 0 (86)

The appropriate tangent scaling factors λ ₀ and λ ₁ can now be solved by first solving for λ ₁ based on λ ₀ and then using that value to arrive at a fourth-order polynomial residue (depressed polynomial): (α 2 / 0β ₀ ) λ 4/0 + (2α ₀ α ₂ β ₀ ) λ 2/0 + (β ₁ α 2/1) λ ₀ + (β ₂ α 2/1 + α 2 / 2β ₀ ) = 0 (87)

Where: α ₀ = 3 / 2κ ₀ (88) α ₁ = -T ₀ × T ₁ (89) α ₂ = -T ₀ × r ₁₀ (90) β ₀ = 3 / 2κ ₁ (91) β ₁ = -T ₀ × T ₁ (92) β ₂ = -T ₁ × r ₁₀ (93)

Heuristic system techniques

A pragmatic approach can be envisaged for solving a cubic G2 curve that has a curvature concentrated at the control points. It is possible to formalize a large nonlinear optimization that simultaneously seeks a solution for all the free parameters that satisfy the equation systems (85) and (86). This approach is expensive and the search space is comparatively large. Instead, a heuristic may be used to estimate the tangent directions and curvatures for the ends of the cubic Bézier curves.
0 tion becomes. So there may be up to four solutions that fulfill the G2 property. Another approach for recognizing this is that equations (85) and (86) form parabolas and that a parsing of these parabolas is sought.
gene vectors is positive. This leads to a problem of determining which of the valid solutions to choose and what to do in particular in a case where there is no solution.

Tangent and curvature assignment

A simple heuristic may be used to match target curves and tangents to the sequence of control points p _i over the radius of the circle. For non-circular points, the quadratic κ-curvature estimation is used. Points that lie in between receive a hybrid of the two curvature values. The process of blending curvature estimates can be termed fairing because it improves the aesthetic appearance of the result.

To accomplish this, a single blending factor (that is, a blend term) can be calculated, which is the product of three separate blending factors, such as a baseline blend, an angle blend, and a bend blend. Each of these factors defines a condition that, in turn, defines where the points in a circular pattern are sufficient to use the circular curvature tangents and curvatures. For each term, the mixing factor is one when the condition is met. Since the final mix factor is a product of all three, the circle estimates are used if all three conditions are sufficiently met. Moreover, a usage control may be a support for each mix factor by raising the factors to a power γ _i . A user can adapt these factors to produce different mixtures. Examples of these conditions are described in detail below.

For the baseline mixing factor, each point r _{i is} projected onto the segment r _-1 and r _{i + 1} . A condition may be established that the projected point p (r _i ) (where p (.) Is the projection operator) is ideally projected onto the midpoint of the segment given by r_ ₁ and r _{i + 1} . If this is so, then the mixed term is 1. If, conversely, the projected point is outside the segment (or at the end points of the segment), then the mixed term drops to zero. Since b is the baseline vector from r _-1 to r _{i + 1} , and let b ⌄ be the normalized form thereof. Then the baseline mixture is given by:

The next term to consider is the angle term. In an ideal configuration, the angle between the two vectors would be r _i r _-1 and r _i r _{i + 1} and exactly equal to 90 °. They are: V ^ ₀ = r _i-1 r _i / | r _i-1 r _i || V ^ ₁ = r _i r _{i + 1} / | r _i r _{i + 1}

The angle mix term becomes: angleBlend = (max (V ^ ₀ × V ^ ₁ , 0)) ^γ1 (95)

With regard to the curve blending term, it has been found in practice that the use of the circular curvature estimation at a given control point makes sense if the adjacent control points have similar circular curvatures. In other words, it should be a sequence of points along the circle. A curvature blending function between two sets of curvatures, which is symmetric, can be defined as follows:

Let ǩ _i-1 , ǩ _i , ǩ _{i + 1} , the circular curvature estimates at r _i-1 , r _i, and r _{i + 1,} respectively. Given the final combined curved blending function given by the product: curvatureBlend (ǩ _i-1 , ǩ _i ) and curvatureBlend (ǩ _i , ǩ _{i + 1} ). Note that if the circular curvatures do not fit on either side of the control point r _i , then the combined angle mixing term is equal to zero.

All three mixed terms can be combined, resulting in a product of four mixing functions. For example, a variety of different choices may be made, such as γ ₀ = 1, γ ₁ = 6 and γ ₂ = 1, but with other choices depending on the taste of the user and how much the user desires Curves of a more circular rounded shape follow, are possible.

Tangent and curvature assignment

After target curvatures and tangent directions are assigned at each control point, the tangent sizes λ ₀ , λ ₁ > 0 are solved using Equation (87). As this is a is residual-polynomial-based (depressed) biquadratic expression, the Ferrari solution can be used directly to provide an analytical solution to the real roots. For each of the real roots, solutions are pruned, where either λ ₀ <0 or λ ₁ <0. Among the remaining solutions, the solution with the largest smallest λ is selected, that is, solutions are preferred in which λ factors are removed from 0.

In some cases, however, an allowable solution can not be found, such as when the tangents are in a "C" configuration, eg, when the resulting curve has a "C" shape. The alternative "S" configuration leads to valid solutions. In the "C" configuration, however, G2 continuity is lost. One technique involves finding a solution that minimizes the deviation of the bends at the two endpoints. This is done directly by minimizing the sum of the squared distances from the desired curvatures, where the solution curvatures satisfy the inequality boundary condition λ ₀ , λ ₁ > 0. In practice, however, running this optimization leads to a solution in which λ ₀ or λ _{1 is} equal to zero.

In another technique, the λ scale factor is fixed at the end of the curve with the smallest curvature size. One or more implementations use one third of the baseline length between the two control points. If the λ-factor is given at one end of the curve, it can be released by directly specifying G2 at the end of the curve that has a greater curvature. This approach hides the G2 discontinuity at the least noticeable end of the curve. This approach is a reasonable choice, as the failures are usually a result of specifying a large curvature at one end of the curve and a small curvature at the other.

As has been explained, it may not always be possible to have G2 continuity for a specified set of tangents, bends, and control points. In addition, there may be a problem that not all G2 solutions are aesthetic and desirable. For example, in some "C" curve configurations, the resulting curves satisfy G2, but the change in curvature at one end is so abrupt that the result approaches a humpy behavior, whereas the resulting curve approaches the oscillating circle near each Endpoint should follow.

Techniques may be employed in one or more implementations that detect such cases and do not allow the λ scale factors to fall below one above the radius of curvature at any end of the curve. When such a case occurs, the curvature is clamped and the λ scaling term is soled to satisfy G2 at the other end of the curve.

Example procedures

The following discussion describes curve techniques that can be used using the systems and devices described above. Aspects of each of the procedures may be implemented in hardware, firmware or software or even a combination thereof. The procedures are presented as a set of blocks that specify operations performed by one or more devices, not necessarily limited to those instructions shown to perform the operations by the respective blocks. In sections of the following discussion will be on 1 to 11 directed.

12 shows a procedure 1200 in an example implementation where a blending factor is used to fit a curve to three or more data points. An indication of three or more data points is received (block 1202 ). The indication may for example be received by a user through an interaction with a user interface, it may be determined from a table, or it may be obtained remotely over a network, and the like.

A blending factor is calculated based on a spatial relationship of the three or more data points to each other (Block 1204 ). The three or more data points may, for example, be arranged in a sequence. This sequence and the spatial relationship of the data points to each other can be used to calculate a mix factor, for example, based on one or more heuristics as described above. Other examples that are based on a spatial relationship alone and not on the sequence are also included.

A curve is fitted to the three or more data points by mixing a plurality of curve fitting techniques using the calculated mixing factor (Block 1206 ). For the example described above, this means that the blending factor can be used to assign weights to different curve fitting techniques and hence curve fitting, and that the techniques used to make this fit can be based, at least in part, on the spatial relationship of the data points.

13 shows a procedure 1300 in an example implementation where a plurality of curve fitting techniques are selected for fitting a curve to a plurality of data points. A choice is made as to which of a plurality of curve fitting techniques to use, based on a spatial relationship of a plurality of data points to which a curve is to be fitted (block 1302 ). The curve fitting module 108 For example, the plurality of data points 110 to determine a curve fitting technique that is likely to result in a curve that matches the desired output of a user. This may be done based on a spatial relationship of the data points, a source of the data points (eg, user input, table, data type), and the like.

The curve is fitted to the plurality of data points using the selected one of the plurality of curve fitting techniques (Block 1304 ). For the previous example, this means that the chosen curve fitting techniques can be used to form the curve. The choice may also include a choice of more than one curve fitting technique, so in such a case these techniques may be mixed together to obtain a final result. A variety of other examples are also included without departing from the spirit and scope thereof.

Exemplary system and device

14 generally indicates 1400 an exemplary system that includes an example computing device 1402 which represents one or more computing systems, and / or devices that can implement the various techniques described herein. This is done by including the curve fitting module 108 which may be configured to fit and / or manipulate curves incorporating data points. The computing device 1402 For example, a server of a service provider, a device associated with a client (eg, a client device), an on-chip system, and / or any other suitable computing device or computing system may be.

The exemplary computing device 1402 as shown includes a processing system 1404 , one or more computer-readable media 1406 and one or more I / O interfaces 1408 that are communicatively coupled with each other. Although not shown, the computing device may 1402 further include a system bus or other data and command transfer system that couples the various components together. A system bus may include any or a combination of different bus structures, such as a memory bus or memory controller, a peripheral bus, a universal serial bus, and / or a processor or local bus using any of a variety of architectures. A variety of such examples are also included, such as control and data lines.

The processing system 1404 provides functionality to perform one or more operations using hardware. Accordingly, the processing system 1404 such that it is a hardware element 1410 which may be more configured as processors, functional blocks, and the like. This may include an implementation in hardware as an application specific integrated circuit or as another logic circuit with formation using one or more semiconductors. The hardware elements 1410 are not limited to materials of which they are formed or to the processing mechanisms used herein. For example, processors may be formed of a semiconductor or semiconductors and / or transistors (eg, electronic integrated circuits (ICs)). In this regard, processor executable instructions may be electronically executable instructions.

The computer-readable storage media 1406 are shown as having a memory / shelf 1412 include. The storage / storage 1412 provides storage / storage capacity associated with one or more computer-readable media. The storage / storage component 1412 For example, volatile media (such as a random access memory (RAM)) and / or nonvolatile media (such as For example, a read only memory (ROM), a flash memory, optical disks, magnetic disks, and the like). The storage / storage component 1412 may include fixed media (eg, RAM, ROM, a hard disk drive, and the like) as well as removable media (eg, a flash memory, a removable drive, an optical disk, and the like). The computer-readable media 1406 can be configured in a variety of other ways as described below.

An input / output interface or input / output interfaces 1408 represent a functionality that allows a user to submit commands and information to the computing device 1402 and also allow information to be presented to the user and / or other components or devices using various input / output devices. Examples of input devices include a keyboard, a cursor control device (eg, a mouse), a microphone, a scanner, touch-sensitive functionality (eg, capacitive or other sensors configured to detect physical motion), a camera (e.g. visible or invisible wavelengths, such as infrared frequencies, may be used to detect movements, such as hand gestures, that do not imply touch), and the like. Examples of output devices include a display device (eg, a monitor or projector), speakers, a printer, a network card, a tactile response device, and the like. The computing device 1402 may therefore be configured in a variety of ways, which will be described below, to assist user interaction.

Various techniques have been described in the general context of software, hardware elements or program modules. In general, such modules include routines, programs, objects, elements, components, data structures, and the like that perform specific tasks or implement specific abstract data types. The terms "module," "functionality," and "component" as used herein refer to software, firmware, hardware, or a combination thereof. The features of the techniques described herein are platform independent, which means that the techniques are compatible with a variety of conventional computing platforms a variety of processors can be implemented.

An implementation of the above-described modules and techniques may be stored on or transmitted through a form of computer-readable media. The computer readable media may include a variety of media to which the computing device 1402 can be accessed. By way of example, and not limitation, computer-readable media may include "computer-readable storage media" and "computer-readable signal media."

"Computer-readable storage media" may refer to media and / or devices that enable persistent and / or non-transient storage of information, as opposed to mere transmission, carrier waves or signals per se. Computer readable storage media refers to non-signal carrier media. The computer readable storage media include hardware, such as volatile and nonvolatile, removable and non-removable media, and / or memory devices implemented in a method or technology capable of storing information, such as computer readable instructions, data structures, program modules, logical elements. Circuits or other data. Include, examples of computer readable storage media, among others, RAM, ROM, EEPROM, flash memory or other memory technology _'CD-ROM, DVD (Digital Versatile Disk) or other optical storage, hard disks, magnetic tapes, magnetic tapes, magnetic disk storage or other magnetic storage devices or other storage device, physical media or object for manufacturing capable of storing the desired information and being accessible by a computer.

"Computer-readable signal media" may refer to a signal carrying medium that is configured to direct instructions to the computing device hardware 1402 for example, to transmit over a network. Signal media may typically embody computer readable instructions, data structures, program modules, or other data in a modulated data signal, such as carrier waves, data signals, or other transport mechanisms. Signal media also contain any information distribution media. The term "modulated data signal" refers to a signal that has one or more characteristics that are chosen or changed to encode information in the signal. By way of example, and not limitation, communication media includes wired media, such as a wired network or a direct wired connection, and wireless media, such as acoustic, wireless, infrared, and other wireless media.

As described above, hardware elements 1410 and computer-readable media 1406 Modules, programmable device logic, and / or fixed device logic with implementation in hardware form, which in some embodiments may be used to implement at least some aspects of the techniques described herein, such as to perform one or more instructions. Hardware may include components of an integrated circuit or on-chip system, an application specific integrated circuit (ASIC), field programmable gate array (FPGA), complex programmable logic device (CPLD) and other implementations in silicon or other hardware. In this regard, hardware may operate as a processing device that performs program tasks defined by instructions and / or hardware embodied logic, as well as hardware used to store instructions for execution, such as the computer readable ones described above storage media.

Combinations of the foregoing may also be used to implement various of the techniques described herein. Accordingly, software, hardware, or executable modules may be embodied as one or more instructions and / or embodying logic in any form of computer-readable storage media and / or by one or more hardware elements 1410 be implemented. The computing device 1402 may be configured to implement certain instructions and / or functions corresponding to the software and / or hardware modules. Accordingly, an implementation of a module may be performed by the computing device 1402 is executable, as software at least partially achieved in hardware, for example by using computer-readable storage media and / or hardware elements 1410 of the processing system 1404 , The instructions and / or functions may be represented by one or more real objects (eg, one or more computing devices 1402 and / or processing systems 1404 ) to implement techniques, modules, and examples as described herein.

The techniques described herein may be achieved by various configurations of the computing device 1402 are not limited to the specific examples of the techniques described herein. This functionality can also be implemented in whole or in part using a distributed system, such as via "cloud" 1414 over a platform 1416 , as will be described below.

The cloud 1414 includes and / or provides a platform 1416 for resources 1418 dar. The platform 1416 abstracts underlying functionality of the hardware (such as servers) and software resources of the cloud 1414 , The resources 1418 may include applications and / or data that are used while executed in computer processing on servers that are hosted by the computing device 1402 are located away. The resources 1418 may also include servers that are accessible via the Internet and / or via a subscriber network, such as a cell-based or a WiFi network.

The platform 1416 can provide resources and functions for connecting the computing device 1402 abstract with other computing devices. The platform 1416 can also serve to abstract the scale of the resources to a corresponding scale of the desired resource requirements 1418 that over the platform 1416 are implemented to provide. Accordingly, in a mutually connected device embodiment, an implementation of the functionality described herein may be via the system 1400 be distributed across. The functionality may be partially on the computing device, for example 1402 over the platform 1416 that the functionality of the cloud 1414 abstracted, implemented.

Mapping techniques for curve fitting are described. In one or more implementations, an indication of three or more data points is received. A blending factor based on a spatial relationship of the three or more data points to each other will be described. A curve is fitted to the three or more data points by mixing a plurality of curve fitting techniques using the calculated blend factor.

concluding remark

While the invention is described in language specific to structural features and / or methodological acts, it should be understood that the invention as defined in the appended claims is not necessarily limited to the specific features or acts described. Rather, the specific features and processes are disclosed as exemplary forms of implementing the claimed invention.

Claims (13)

A method implemented by one or more computing devices, the method comprising: Receiving an indication of three or more data points; Calculating a blend factor based on a spatial relationship of the three or more data points to each other; and Fitting a curve to the three or more data points by mixing a plurality of curve fitting techniques using the calculated blend factor. The method of claim 1, wherein the plurality of curve fitting techniques include square Bezier curve techniques, cubic Bezier curve techniques, and circular curve estimation techniques. The method of any one of the preceding claims, wherein said fitting includes calculating a tangent direction and a radius of curvature; and / or wherein the spatial relationship describes a symmetry of the three or more points to each other, wherein the symmetry of the three or more data points to each other is preferably based on whether the successively defined first, second and third data points, the second data point between one of the first and third data points defined baseline falls. The method of claim 1, wherein the spatial relationship describes a flatness of the three or more data points to each other, wherein the flatness of the three or more data points relative to one another preferably at a proximity given for the successively defined first, second, and third data points second data point is based on a baseline defined by the first and third data points. The method of any one of the preceding claims, wherein the spatial relationship is based at least in part on a curvature sign of the adjacent data point, the curvature sign preferably being used by determining whether the curvature sign of the adjacent data point is different from a sign for the current data point. The method of any one of the preceding claims, wherein the calculating is performed for a window defining a number of data points positioned before and after the current data point. A computer program product comprising computer readable instructions which, when loaded into and executed on a suitable system, perform the steps of a method according to any one of the preceding claims. One or more computer-readable storage media having instructions stored thereon that, in response to execution by a computing device, cause the computing device to perform operations that include: Choosing which of a plurality of curve fitting techniques to apply, based on a spatial relationship of a plurality of data points to which a curve is to be fitted; and Fitting the curve to the plurality of data points using the selected one of the plurality of curve fitting techniques. The one or more computer readable storage media of claim 8, wherein the selecting includes calculating an estimate using a quadratic Bezier curve technique, and the plurality of curve fitting techniques include cubic Bezier curve techniques and circular curve estimation techniques; and / or wherein the spatial relationship is described using a heuristic that includes symmetry or flatness of the plurality of data points; and / or wherein the spatial relationship is described using a heuristic that includes whether a curvature sign of the adjacent data point is different from a curvature sign at the current data point. A system comprising: one or more modules that are at least partially implemented in hardware, wherein the one or more modules are configured to perform operations including: Receiving an indication to adjust a curve to a plurality of data points; Determining a spatial relationship of the plurality of data points to each other; and mixing a plurality of curve fitting techniques to perform the fitting of the curve, the mixing using at least one heuristic based on the determined spatial relationship. The system of claim 10, wherein the plurality of curve fitting techniques includes quadratic Bézier curve techniques, cubic Bezier curve techniques, and circular curve estimation techniques; and / or wherein the heuristic employs a quadratic Bézier curve technique; and / or wherein the heuristic is based on symmetry of the plurality of data points based on whether, for the first, second and third data points defined in succession, the second data point falls between a baseline defined by the first and third data points; and / or wherein the heuristic is based on a flatness of the plurality of data points to each other based on a proximity given to the sequentially-defined first, second, and third data points with which the second data point falls on a baseline defined by the first and third data points; and / or wherein the heuristic is based on a determination of whether a curvature sign of the adjacent data point is different from a sign for the current data point. System comprising: One or more modules implemented at least in part in hardware, wherein the one or more modules are configured to perform operations including: Configuring contact points located directly on a complex curve in a user interface to support a user specified motion; and in response to the user specified movement with respect to a respective contact point, automatically and without user intervention, adjusting the respective curve of the complex curve such that the respective curve retains the slope and curvature continuity with respect to the complex curve. The system of claim 12, wherein the contact points are maintained according to the movement specified by the user through the movement; and / or wherein the adjustment is carried out in such a way that, following the movement, the contact point affected during the movement is close to a position of the maximum curvature of the respective curve.

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